Question 2 (10 points) Evaluate \( \int_{1}^{e} x^{49} \ln x d x \)

To evaluate the integral \( I = \int_{1}^{e} x^{49} \ln x \, dx \), we can use integration by parts. Let: - \( u = \ln x \) then \( du = \frac{1}{x} \, dx \) - \( dv = x^{49} \, dx \) then \( v = \frac{x^{50}}{50} \) Applying integration by parts, we have: \[ I = \left[ \frac{x^{50}}{50} \ln x \right]_{1}^{e} - \int_{1}^{e} \frac{x^{50}}{50} \cdot \frac{1}{x} \, dx \] Calculating the boundary term: At \( x = e \): \[ \frac{e^{50}}{50} \ln e = \frac{e^{50}}{50} \cdot 1 = \frac{e^{50}}{50} \] At \( x = 1 \): \[ \frac{1^{50}}{50} \ln 1 = \frac{1}{50} \cdot 0 = 0 \] Thus, the boundary term evaluates to \( \frac{e^{50}}{50} - 0 = \frac{e^{50}}{50} \). Now, we need to evaluate the remaining integral: \[ \int_{1}^{e} \frac{x^{50}}{50} \cdot \frac{1}{x} \, dx = \int_{1}^{e} \frac{x^{50}}{50} \, dx = \frac{1}{50} \int_{1}^{e} x^{49} \, dx \] Calculating this integral: \[ \int x^{49} \, dx = \frac{x^{50}}{50} \Big|_{1}^{e} = \frac{e^{50}}{50} - \frac{1^{50}}{50} = \frac{e^{50}}{50} - \frac{1}{50} \] Thus, we have: \[ \int_{1}^{e} x^{49} \, dx = \frac{e^{50} - 1}{50} \] Putting this back into our expression for \( I \): \[ I = \frac{e^{50}}{50} - \frac{1}{50} \cdot \frac{1}{50} (e^{50} - 1) = \frac{e^{50}}{50} - \frac{e^{50} - 1}{2500} \] Now simplifying this: \[ I = \frac{e^{50}}{50} - \frac{e^{50}}{2500} + \frac{1}{2500} = \frac{50e^{50}}{2500} - \frac{e^{50}}{2500} + \frac{1}{2500} = \frac{49e^{50} + 1}{2500} \] Thus, the final answer is: \[ \int_{1}^{e} x^{49} \ln x \, dx = \frac{49e^{50} + 1}{2500} \]